Definition 27.4.1. Let $P$ be a property of rings. We say that $P$ is local if the following hold:

1. For any ring $R$, and any $f \in R$ we have $P(R) \Rightarrow P(R_ f)$.

2. For any ring $R$, and $f_ i \in R$ such that $(f_1, \ldots , f_ n) = R$ then $\forall i, P(R_{f_ i}) \Rightarrow P(R)$.

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